Rationality of Euler–Chow series and finite generation of Cox rings

Abstract

We study the rationality of the Euler–Chow series E1(X) of codimension one cycles on a projective variety X and its relation with the effective cone and the Cox ring of X. Among other things, we prove that E1(X) is transcendental if the cone NE1(X) of pseudo-effective divisors on X has infinitely many extremal rays generated by effective divisors. On the other hand, we give examples showing that the converse fails. In addition, we give an example where E1(X) is rational and Cox(X) is infinitely generated. Finally, we compute E1(X) for Del Pezzo surfaces X.